WASHINGTON — Scott Brown’s expected run for US Senate from New Hampshire has helped boost both Republicans’ confidence and their chances, analysts say, of recapturing the chamber.

Prognosticators are now putting GOP odds of winning control of the Senate at 50-50 or better, a marked improvement from just a few weeks ago. With continued Republican control of the House expected, the loss of the Senate for Democrats — after holding the majority for eight years — would make President Obama’s political hand even weaker during his final two years in office.

Brown’s announcement that he would begin touring the state and raising money for a likely run against Democratic incumbent Jeanne Shaheen adds New Hampshire to a growing roster of up to 11 states where Republicans are increasingly on the offensive, forcing Democrats to spread their resources to defend their turf.

“It adds to the playing field,’’ said Ron Kaufman, a Republican strategist and fund-raiser from Massachusetts who said Brown has a strong shot at winning. “It makes it harder for the Democrats to defend their folks who are in trouble.’’

Brown begins his campaign activities with recent polls showing Shaheen leading by 9 percent to 13 percent. But negative ads run by outside groups against Shaheen, hammering her support for Obama’s health care law, are taking a toll. Her unfavorable rating climbed to 34 percent in January, up from 22 percent three months earlier, according to the Granite State Poll, which is sponsored by WMUR-TV and conducted by the University of New Hampshire.

Some outside political groups began buying ads after Brown’s Friday announcement that he was taking a key step toward entering the race.

Democrats and Republicans are both predicting that candidates and outside groups could spend more than $20 million on the New Hampshire race, which would make it the state’s most expensive ever. Many of those campaign ads will run on Boston television stations, which broadcast to Southern New Hampshire.

Still, the sentiment among national Republicans is not unfettered excitement over Brown’s candidacy, at least not of the sort that swept the GOP in 2010 when the relatively unknown state senator captured the late Edward M. Kennedy’s Senate seat in a special election. Brown lost the seat to Elizabeth Warren in 2012.

This time around, as he prepares to run over the Bay State’s northern border, the Republicans know he is an underdog but see his candidacy as providing at minimum an additional tactical advantage. They hope his bid will help suck money away from the Democratic efforts elsewhere.

“It takes another race that will at the very least be competitive — with a chance to win — and all of a sudden you have Democrats having to move resources,’’ said John Brabender, a national Republican consultant. “It complicates things for the Democrats.”

Republicans need to gain six seats to reclaim the majority in the upper chamber. At least two seats currently held by Democrats — in South Dakota and West Virginia — are already considered likely to flip to Republican hands.

Previously, the GOP had been most closely focused on Democratic-held seats in North Carolina, Montana, Arkansas, Louisiana, and Alaska, states that Obama lost in 2012. That roster of potential pickoffs left the Republicans little margin for error.

But Republicans in recent weeks have expanded their prospects. In addition to Brown’s candidacy, US Representative Cory Gardner, a Colorado Republican, said three weeks ago that he would challenge incumbent Senator Mark Udall, a Democrat who previously had an easy path to reelection. Races for Democratic-held seats in Michigan and Iowa also have grown more competitive for Republicans as polling has shifted.

“The environment overall is just very good for them right now,” said Jennifer Duffy, senior editor for the nonpartisan Cook Political Report, which rates congressional races across the country.

While Republicans are seeing numerous opportunities, there are almost no chances for Democrats to pick up a seat held by a Republican. Analysts mention Kentucky and Georgia, but these are Southern states where President Obama is deeply unpopular.

A president’s party almost always loses Senate seats in midterm elections. Republicans lost six seats in 2006 under George W. Bush and eight seats in 1986 under Ronald Reagan. Bill Clinton was the only recent anomaly, with Democrats not losing any seats in 1998.

But recent polls have given Obama the lowest approval ratings of his presidency, and there has been little indication that Americans are warming to his signature health care plan.

In a state where some rural residents — because of new, limited insurance networks — have had to drive longer distances to receive certain types of care, Shaheen is particularly vulnerable on health care. The state is among those that has refused to offer expanded Medicaid coverage under the new law, although the state Legislature is considering whether to offer such an expansion.

The Cook Political Report has upgraded odds of a Republican takeover from 40 percent to 50 percent in recent weeks, and says the GOP’s odds will grow larger if the most electable candidates survive party primaries.

Stu Rothenberg, another nonpartisan handicapper, said he agrees the odds for Republicans are 50 percent or better. He added there is an “ominous outlook for Democrats right now.”

The New Hampshire race “begins as an interesting Republican opportunity that a week ago didn’t exist,” Rothenberg said. He also pointed to Republican excitement over last week’s special election in a House race in Florida, where the GOP candidate defeated a well-funded Democrat in a toss-up district by emphasizing Obama’s health law.

Democrats showed their anxiety over the weekend with a flood of e-mails to supporters trying to denigrate Brown and raise money for Shaheen. Senator Elizabeth Warren, a powerful fund-raiser who has raised more for Shaheen than any other candidate, sent an e-mail to her supporters hours after Brown opened his exploratory committee on Friday.

Democrats also tried to pressure Brown to sign a pledge with Shaheen limiting the ability of outside groups to spend money in the race, similar to one he signed with Warren in the 2012 election. That pledge was widely seen as benefiting Warren, in part because it did not limit the ability of unions to organize effective get-out-the-vote operations that traditionally benefit Democrats.

One indication that Democrats are already thinking about their resources: The Democratic Senatorial Campaign Committee is not planning to upgrade New Hampshire to one of its seven premier races where it intends to spend the most money, according to a top Democratic official. That could change if the race tightens.

In heavily Democratic Massachusetts in 2012, Democrats emphasized that a Brown victory over Warren could put the Senate in Republican hands. But in New Hampshire, a toss-up state where Obama is not popular, the party will try to instead focus on the candidates.

Even before Brown made his announcement Friday, Democrats have been reminding voters that he ran less than two years ago as a sitting senator from Massachusetts, with an emphasis on his local ties.

Outside groups have not wasted any time. The League of Conservation Voters spent $200,000 on anti-Brown ads in February, criticizing his votes in favor of retaining oil subsidies.

American Crossroads, a Republican super PAC, is already launching a $600,000 ad campaign against Shaheen this week. That comes on top of about $2.4 million that Americans for Prosperity, a conservative nonprofit, has spent over the past year.

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